Theorem elsnxp 6322

Description: Membership in a Cartesian product with a singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.) (Proof shortened by JJ, 14-Jul-2021.)

Assertion

Ref	Expression
elsnxp	⊢ (𝑋 ∈ 𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩))

Distinct variable groups:   𝑦,𝐴   𝑦,𝑉   𝑦,𝑋   𝑦,𝑍

Proof of Theorem elsnxp

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp 5723	. . 3 ⊢ (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑥∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)))
2		df-rex 3077	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
3		an13 646	. . . . . . 7 ⊢ ((𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) ↔ (𝑦 ∈ 𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
4	3	exbii 1846	. . . . . 6 ⊢ (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
5	2, 4	bitr4i 278	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)))
6		elsni 4665	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑋} → 𝑥 = 𝑋)
7	6	opeq1d 4903	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑋} → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
8	7	eqeq2d 2751	. . . . . . 7 ⊢ (𝑥 ∈ {𝑋} → (𝑍 = ⟨𝑥, 𝑦⟩ ↔ 𝑍 = ⟨𝑋, 𝑦⟩))
9	8	biimpa 476	. . . . . 6 ⊢ ((𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
10	9	reximi 3090	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
11	5, 10	sylbir 235	. . . 4 ⊢ (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
12	11	exlimiv 1929	. . 3 ⊢ (∃𝑥∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
13	1, 12	sylbi 217	. 2 ⊢ (𝑍 ∈ ({𝑋} × 𝐴) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
14		snidg 4682	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
15		opelxpi 5737	. . . . 5 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
16	14, 15	sylan 579	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
17		eleq1 2832	. . . 4 ⊢ (𝑍 = ⟨𝑋, 𝑦⟩ → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴)))
18	16, 17	syl5ibrcom 247	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → (𝑍 = ⟨𝑋, 𝑦⟩ → 𝑍 ∈ ({𝑋} × 𝐴)))
19	18	rexlimdva 3161	. 2 ⊢ (𝑋 ∈ 𝑉 → (∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩ → 𝑍 ∈ ({𝑋} × 𝐴)))
20	13, 19	impbid2 226	1 ⊢ (𝑋 ∈ 𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076  {csn 4648  ⟨cop 4654   × cxp 5698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-xp 5706

This theorem is referenced by:  esum2dlem  34056  esum2d  34057  projf1o  45104  sge0xp  46350